Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $t \neq 0$. $p = \dfrac{-t + 8}{t - 4} \times \dfrac{t^2 + 4t - 32}{-t^2 - 4t + 32} $
Solution: First factor out any common factors. $p = \dfrac{-(t - 8)}{t - 4} \times \dfrac{t^2 + 4t - 32}{-(t^2 + 4t - 32)} $ Then factor the quadratic expressions. $p = \dfrac {-(t - 8)} {t - 4} \times \dfrac {(t + 8)(t - 4)} {-(t + 8)(t - 4)} $ Then multiply the two numerators and multiply the two denominators. $p = \dfrac {-(t - 8) \times (t + 8)(t - 4) } {(t - 4) \times -(t + 8)(t - 4) } $ $p = \dfrac {-(t + 8)(t - 4)(t - 8)} {-(t + 8)(t - 4)(t - 4)} $ Notice that $(t + 8)$ and $(t - 4)$ appear in both the numerator and denominator so we can cancel them. $p = \dfrac {-\cancel{(t + 8)}(t - 4)(t - 8)} {-\cancel{(t + 8)}(t - 4)(t - 4)} $ We are dividing by $t + 8$ , so $t + 8 \neq 0$ Therefore, $t \neq -8$ $p = \dfrac {-\cancel{(t + 8)}\cancel{(t - 4)}(t - 8)} {-\cancel{(t + 8)}(t - 4)\cancel{(t - 4)}} $ We are dividing by $t - 4$ , so $t - 4 \neq 0$ Therefore, $t \neq 4$ $p = \dfrac {-(t - 8)} {-(t - 4)} $ $ p = \dfrac{t - 8}{t - 4}; t \neq -8; t \neq 4 $